Archdeacon Pratt on Attractions. 445 



is required is, that they should not die during that season. 

 But all botanists know perfectly well that a forest-tree would 

 not perish simply in consequence of being deprived of the 

 direct rays of the sun for a few months, provided that it 

 enjoyed a sufficient amount of heat. We are, however, not 

 even warranted to conclude that the absence of the sun would 

 prevent growth. It would affect the colour of the leaves ; but 

 it is doubtful whether it would seriously affect the growth of 

 the tree. From a paper read before the French Academy, 

 April 9, 1866, by M. P. Duchartre, it would appear that some 

 plants grow much faster during the night, when deprived of the 

 sun's rays, than during daylight. 



LVI. On the Problem on Attractions, in the Philosophical Ma- 

 gazine, page 332. By Archdeacon Pratt, M.A., F.R.S. 



To the Editors of the Philosophical Magazine and Journal, 

 Gentlemen, 



THE latter part of the solution I sent you, March 7, 1867, 

 may also be done as follows : — 

 I have shown that the condition 



(*1 /»2ir u i+3 f D (l) ~] 



1,1 7 <ITs{ pw -^ +&c -} dlid "= 



must be satisfied, i being any whole number, and u, p (0) , p^ . . . 

 functions of p, and co, and independent of i ; and P; Laplace's 

 coefficient of the ith order, the same function of pJ and co' (the 

 coordinates of the external attracted point) as of p, and co. Put 

 the whole expression under the signs of integration, except P ? , 

 equal to cf> (p,, co) = cf) + cp l + ... a series of Laplace's functions. 

 Then by a property of those functions*, 



'2tt 4 <7r 



Y { fap,, to) dfi dco = ^-—^ <j>' it 



u: 



where fai is the same function of /J and co' that fa is of p, and co. 

 But this expression equals zero by the condition. Hence fai = 

 and also cj) i =0 for all integral values of i; 



S {><•>- fi +••■•>-* 



where fa is independent of p, and co, and therefore of u, and a 

 function of i only. This must be true, however large i is. Now 

 pW, the density at the surface, must always be some finite quan- 

 tity. Consequently, as the condition now stands, when i is taken 

 * See iny ' Figure of the Earth/ 3rd. edit. p. 29. 



